Would this horse be good as my first?

First off you sound over confident which can be a very dangerous thing, you are only 13 and you don't know that much yet. World famous horse trainer Monty Roberts even at his old age and a whole lifetime as a horse trainer often says that he's still has a lot to learn. You can't be able to diagnose lameness unless you've had formal training, which at 13 you haven't. That being said you are probably are at a point where owning your own horse could benefit you. You've rode plenty of other horses but owning a horse changes a lot of things. Being able to ride different horses is good but learning to work into a deeper working relationship with a horse is also important which horse ownership can hopefully teach you. This mare sounds trained well enough to bring you as a rider to the next level, she is the right age, and if you will be working with a trainer once a week and riding her 3 to 5 days a week it should work out. With work and dedication she can win you ribbons. She is also just tall enough to be used for years without risk of outgrowing her. Here in the USA most people try to get a two week trial period to make sure the horse will work out. If you can talk them into doing a two week trail then your mother will also have a chance to decide if this horse is right for you. Since your parents almost bought you a horse it just sounds like they just want you to have a horse that will be good for you.